gva.augmentedRCBD performs genetic variability analysis on an object of class augmentedRCBD.

gva.augmentedRCBD(aug, k = 2.063)

Arguments

aug

An object of class augmentedRCBD.

k

The standardized selection differential or selection intensity. Default is 2.063 for 5% selection proportion (see Details).

Value

A list with the following descriptive statistics:

Mean

The mean value.

PV

Phenotyic variance.

GV

Genotyipc variance.

EV

Environmental variance.

GCV

Genotypic coefficient of variation

GCV category

The GCV category according to Sivasubramaniam and Madhavamenon (1973) .

PCV

Phenotypic coefficient of variation

PCV category

The PCV category according to Sivasubramaniam and Madhavamenon (1973) .

ECV

Environmental coefficient of variation

hBS

The broad-sense heritability (H2) (Lush 1940) .

hBS category

The H2 category according to Robinson (1966) .

GA

Genetic advance (Johnson et al. 1955) .

GAM

Genetic advance as per cent of mean (Johnson et al. 1955) .

GAM category

The GAM category according to Johnson et al. (1955) .

Details

gva.augmentedRCBD performs genetic variability analysis from the ANOVA results in an object of class augmentedRCBD and computes several variability estimates.

The phenotypic, genotypic and environmental variance (σ2p, σ2g and σ2e ) are obtained from the ANOVA tables according to the expected value of mean square described by Federer and Searle (1976) as follows:

σ2p = Mean sum of squares of test treatments

σ2e = Mean sum of squares of residuals

σ2g = σ2pσ2e

Phenotypic and genotypic coefficients of variation (PCV and GCV) are estimated according to Burton (1951, 1952) as follows:

PCV = [σ2p√ (x)] × 100

GCV = [σ2g√ (x)] × 100

Where x is the mean.

The estimates of PCV and GCV are categorised according to Sivasubramanian and Madhavamenon (1978) as follows:

CV (%)Category
x < 10Low
10 ≤ x < 20Medium
≥ 20High

The broad-sense heritability (H2) is calculated according to method of Lush (1940) as follows:

H2 = σ2gσ2p

The estimates of broad-sense heritability (H2) are categorised according to Robinson (1966) as follows:

H2Category
x < 30Low
30 ≤ x < 60Medium
≥ 60High

Genetic advance (GA) is estimated and categorised according to Johnson et al., (1955) as follows:

GA = k × σg × [H2 ⁄ 100]

Where the constant k is the standardized selection differential or selection intensity. The value of k at 5% proportion selected is 2.063. Values of k at other selected proportions are available in Appendix Table A of Falconer and Mackay (1996).

Selection intensity (k) can also be computed in R as below:

If p is the proportion of selected individuals, then deviation of truncation point from mean (x) and selection intensity (k) are as follows:

x = qnorm(1-p)

k = dnorm(qnorm(1 - p))/p

Using the same the Appendix Table A of Falconer and Mackay (1996) can be recreated as follows.

TableA <- data.frame(p = c(seq(0.01, 0.10, 0.01), NA,
                           seq(0.10, 0.50, 0.02), NA,
                           seq(1, 5, 0.2), NA,
                           seq(5, 10, 0.5), NA,
                           seq(10, 50, 1)))
TableA$x <- qnorm(1-(TableA$p/100))
TableA$i <- dnorm(qnorm(1 - (TableA$p/100)))/(TableA$p/100)

Appendix Table A (Falconer and Mackay, 1996)

p%xi
0.013.719016493.9584797
0.023.540083803.7892117
0.033.431614403.6869547
0.043.352794783.6128288
0.053.290526733.5543807
0.063.238880123.5059803
0.073.194651053.4645890
0.083.155906763.4283756
0.093.121389153.3961490
0.103.090232313.3670901
<><><>
0.103.090232313.3670901
0.123.035672373.3162739
0.142.988882273.2727673
0.162.947842553.2346647
0.182.911237733.2007256
0.202.878161743.1700966
0.222.847963293.1421647
0.242.820158063.1164741
0.262.794375873.0926770
0.282.770327233.0705013
0.302.747781393.0497304
0.322.726551323.0301887
0.342.706483313.0117321
0.362.687449452.9942406
0.382.669342092.9776133
0.402.652069812.9617646
0.422.635554242.9466212
0.442.619727712.9321196
0.462.604531362.9182048
0.482.589913682.9048286
0.502.575829302.8919486
<><><>
1.002.326347872.6652142
1.202.257129242.6028159
1.402.197286382.5490627
1.602.144410622.5017227
1.802.096927432.4593391
2.002.053748912.4209068
2.202.014090812.3857019
2.401.977368432.3531856
2.601.943133752.3229451
2.801.911035652.2946575
3.001.880793612.2680650
3.201.852179862.2429584
3.401.825006822.2191656
3.601.799118112.1965431
3.801.774381912.1749703
4.001.750686072.1543444
4.201.727934322.1345772
4.401.706043402.1155928
4.601.684940772.0973249
4.801.664562862.0797152
5.001.644853632.0627128
<><><>
5.001.644853632.0627128
5.501.598193142.0225779
6.001.554773591.9853828
6.501.514101891.9506784
7.001.475791031.9181131
7.501.439531471.8874056
8.001.405071561.8583278
8.501.372203811.8306916
9.001.340755031.8043403
9.501.310579111.7791417
10.001.281551571.7549833
<><><>
10.001.281551571.7549833
11.001.226528121.7094142
12.001.174986791.6670040
13.001.126391131.6272701
14.001.080319341.5898336
15.001.036433391.5543918
16.000.994457881.5206984
17.000.954165251.4885502
18.000.915365091.4577779
19.000.877896301.4282383
20.000.841621231.3998096
21.000.806421251.3723871
22.000.772193211.3458799
23.000.738846851.3202091
24.000.706302561.2953050
25.000.674489751.2711063
26.000.643345411.2475585
27.000.612812991.2246130
28.000.582841511.2022262
29.000.553384721.1803588
30.000.524400511.1589754
31.000.495850351.1380436
32.000.467698801.1175342
33.000.439913171.0974204
34.000.412463131.0776774
35.000.385320471.0582829
36.000.358458791.0392158
37.000.331853351.0204568
38.000.305480791.0019882
39.000.279319030.9837932
40.000.253347100.9658563
41.000.227544980.9481631
42.000.201893480.9306998
43.000.176374160.9134539
44.000.150969220.8964132
45.000.125661350.8795664
46.000.100433720.8629028
47.000.075269860.8464123
48.000.050153580.8300851
49.000.025068910.8139121
50.000.000000000.7978846

Where p% is the selected percentage of individuals from a population, x is the deviation of the point of truncation of selected individuals from population mean and i is the selection intensity.

Genetic advance as per cent of mean (GAM) are estimated and categorised according to Johnson et al., (1955) as follows:

GAM = [ GAx ] × 100

GAMCategory
x < 10Low
10 ≤ x < 20Medium
≥ 20High

Note

Genetic variability analysis needs to be performed only if the sum of squares of "Treatment: Test" are significant.

Negative estimates of variance components if computed are not abnormal. For information on how to deal with these, refer Dudley and Moll (1969).

References

Lush JL (1940). “Intra-sire correlations or regressions of offspring on dam as a method of estimating heritability of characteristics.” Proceedings of the American Society of Animal Nutrition, 1940(1), 293--301.

Burton GW (1951). “Quantitative inheritance in pearl millet (Pennisetum glaucum).” Agronomy Journal, 43(9), 409--417.

Burton GW (1952). “Qualitative inheritance in grasses. Vol. 1.” In Proceedings of the 6th International Grassland Congress, Pennsylvania State College, 17--23.

Johnson HW, Robinson HF, Comstock RE (1955). “Estimates of genetic and environmental variability in soybeans.” Agronomy journal, 47(7), 314--318.

Robinson HF (1966). “Quantitative genetics in relation to breeding on centennial of Mendelism.” Indian Journal of Genetics and Plant Breeding, 171.

Dudley JW, Moll RH (1969). “Interpretation and use of estimates of heritability and genetic variances in plant breeding.” Crop Science, 9(3), 257--262.

Sivasubramaniam S, Madhavamenon P (1973). “Genotypic and phenotypic variability in rice.” The Madras Agricultural Journal, 60(9-13), 1093--1096.

Federer WT, Searle SR (1976). “Model Considerations and Variance Component Estimation in Augmented Completely Randomized and Randomized Complete Blocks Designs-Preliminary Version.” Technical Report BU-592-M, Cornell University, New York.

Falconer DS, Mackay TFC (1996). Introduction to quantitative genetics. Pearson/Prenctice Hall, New York, NY.

See also

Examples

# Example data blk <- c(rep(1,7),rep(2,6),rep(3,7)) trt <- c(1, 2, 3, 4, 7, 11, 12, 1, 2, 3, 4, 5, 9, 1, 2, 3, 4, 8, 6, 10) y1 <- c(92, 79, 87, 81, 96, 89, 82, 79, 81, 81, 91, 79, 78, 83, 77, 78, 78, 70, 75, 74) y2 <- c(258, 224, 238, 278, 347, 300, 289, 260, 220, 237, 227, 281, 311, 250, 240, 268, 287, 226, 395, 450) data <- data.frame(blk, trt, y1, y2) # Convert block and treatment to factors data$blk <- as.factor(data$blk) data$trt <- as.factor(data$trt) # Results for variable y1 out1 <- augmentedRCBD(data$blk, data$trt, data$y1, method.comp = "lsd", alpha = 0.05, group = TRUE, console = TRUE)
#> #> Augmented Design Details #> ======================== #> #> Number of blocks "3" #> Number of treatments "12" #> Number of check treatments "4" #> Number of test treatments "8" #> Check treatments "1, 2, 3, 4" #> #> ANOVA, Treatment Adjusted #> ========================= #> Df Sum Sq Mean Sq F value Pr(>F) #> Block (ignoring Treatments) 2 360.1 180.04 6.675 0.0298 * #> Treatment (eliminating Blocks) 11 285.1 25.92 0.961 0.5499 #> Treatment: Check 3 52.9 17.64 0.654 0.6092 #> Treatment: Test and Test vs. Check 8 232.2 29.02 1.076 0.4779 #> Residuals 6 161.8 26.97 #> --- #> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 #> #> ANOVA, Block Adjusted #> ===================== #> Df Sum Sq Mean Sq F value Pr(>F) #> Treatment (ignoring Blocks) 11 575.7 52.33 1.940 0.215 #> Treatment: Check 3 52.9 17.64 0.654 0.609 #> Treatment: Test 7 505.9 72.27 2.679 0.125 #> Treatment: Test vs. Check 1 16.9 16.87 0.626 0.459 #> Block (eliminating Treatments) 2 69.5 34.75 1.288 0.342 #> Residuals 6 161.8 26.97 #> #> Treatment Means #> =============== #> Treatment Block Means SE r Min Max Adjusted Means #> 1 1 84.66667 3.844188 3 79 92 84.66667 #> 2 10 3 74.00000 NA 1 74 74 77.25000 #> 3 11 1 89.00000 NA 1 89 89 86.50000 #> 4 12 1 82.00000 NA 1 82 82 79.50000 #> 5 2 79.00000 1.154701 3 77 81 79.00000 #> 6 3 82.00000 2.645751 3 78 87 82.00000 #> 7 4 83.33333 3.929942 3 78 91 83.33333 #> 8 5 2 79.00000 NA 1 79 79 78.25000 #> 9 6 3 75.00000 NA 1 75 75 78.25000 #> 10 7 1 96.00000 NA 1 96 96 93.50000 #> 11 8 3 70.00000 NA 1 70 70 73.25000 #> 12 9 2 78.00000 NA 1 78 78 77.25000 #> #> Coefficient of Variation #> ======================== #> 6.372367 #> #> Overall Adjusted Mean #> ===================== #> 81.0625 #> #> Standard Errors #> =================== #> Std. Error of Diff. CD (5%) #> Control Treatment Means 4.240458 10.37603 #> Two Test Treatments (Same Block) 7.344688 17.97180 #> Two Test Treatments (Different Blocks) 8.211611 20.09309 #> A Test Treatment and a Control Treatment 6.704752 16.40594 #> #> Treatment Groups #> ================== #> #> Method : lsd #> #> Treatment Adjusted Means SE df lower.CL upper.CL Group #> 8 8 73.25000 5.609598 6 59.52381 86.97619 1 #> 9 9 77.25000 5.609598 6 63.52381 90.97619 12 #> 10 10 77.25000 5.609598 6 63.52381 90.97619 12 #> 5 5 78.25000 5.609598 6 64.52381 91.97619 12 #> 6 6 78.25000 5.609598 6 64.52381 91.97619 12 #> 2 2 79.00000 2.998456 6 71.66304 86.33696 12 #> 12 12 79.50000 5.609598 6 65.77381 93.22619 12 #> 3 3 82.00000 2.998456 6 74.66304 89.33696 12 #> 4 4 83.33333 2.998456 6 75.99637 90.67029 12 #> 1 1 84.66667 2.998456 6 77.32971 92.00363 12 #> 11 11 86.50000 5.609598 6 72.77381 100.22619 12 #> 7 7 93.50000 5.609598 6 79.77381 107.22619 2
# Results for variable y2 out2 <- augmentedRCBD(data$blk, data$trt, data$y2, method.comp = "lsd", alpha = 0.05, group = TRUE, console = TRUE)
#> #> Augmented Design Details #> ======================== #> #> Number of blocks "3" #> Number of treatments "12" #> Number of check treatments "4" #> Number of test treatments "8" #> Check treatments "1, 2, 3, 4" #> #> ANOVA, Treatment Adjusted #> ========================= #> Df Sum Sq Mean Sq F value Pr(>F) #> Block (ignoring Treatments) 2 7019 3510 12.261 0.007597 ** #> Treatment (eliminating Blocks) 11 58965 5360 18.727 0.000920 *** #> Treatment: Check 3 2150 717 2.504 0.156116 #> Treatment: Test and Test vs. Check 8 56815 7102 24.810 0.000473 *** #> Residuals 6 1717 286 #> --- #> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 #> #> ANOVA, Block Adjusted #> ===================== #> Df Sum Sq Mean Sq F value Pr(>F) #> Treatment (ignoring Blocks) 11 64708 5883 20.550 0.000707 *** #> Treatment: Check 3 2150 717 2.504 0.156116 #> Treatment: Test 7 34863 4980 17.399 0.001366 ** #> Treatment: Test vs. Check 1 27694 27694 96.749 6.36e-05 *** #> Block (eliminating Treatments) 2 1277 639 2.231 0.188645 #> Residuals 6 1718 286 #> --- #> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 #> #> Treatment Means #> =============== #> Treatment Block Means SE r Min Max Adjusted Means #> 1 1 256.0000 3.055050 3 250 260 256.0000 #> 2 10 3 450.0000 NA 1 450 450 437.6667 #> 3 11 1 300.0000 NA 1 300 300 299.4167 #> 4 12 1 289.0000 NA 1 289 289 288.4167 #> 5 2 228.0000 6.110101 3 220 240 228.0000 #> 6 3 247.6667 10.170764 3 237 268 247.6667 #> 7 4 264.0000 18.681542 3 227 287 264.0000 #> 8 5 2 281.0000 NA 1 281 281 293.9167 #> 9 6 3 395.0000 NA 1 395 395 382.6667 #> 10 7 1 347.0000 NA 1 347 347 346.4167 #> 11 8 3 226.0000 NA 1 226 226 213.6667 #> 12 9 2 311.0000 NA 1 311 311 323.9167 #> #> Coefficient of Variation #> ======================== #> 6.057617 #> #> Overall Adjusted Mean #> ===================== #> 298.4792 #> #> Standard Errors #> =================== #> Std. Error of Diff. CD (5%) #> Control Treatment Means 13.81424 33.80224 #> Two Test Treatments (Same Block) 23.92697 58.54719 #> Two Test Treatments (Different Blocks) 26.75117 65.45775 #> A Test Treatment and a Control Treatment 21.84224 53.44603 #> #> Treatment Groups #> ================== #> #> Method : lsd #> #> Treatment Adjusted Means SE df lower.CL upper.CL Group #> 8 8 213.6667 18.274527 6 168.9505 258.3828 12 #> 2 2 228.0000 9.768146 6 204.0982 251.9018 1 #> 3 3 247.6667 9.768146 6 223.7649 271.5685 123 #> 1 1 256.0000 9.768146 6 232.0982 279.9018 1234 #> 4 4 264.0000 9.768146 6 240.0982 287.9018 234 #> 12 12 288.4167 18.274527 6 243.7005 333.1328 345 #> 5 5 293.9167 18.274527 6 249.2005 338.6328 345 #> 11 11 299.4167 18.274527 6 254.7005 344.1328 45 #> 9 9 323.9167 18.274527 6 279.2005 368.6328 56 #> 7 7 346.4167 18.274527 6 301.7005 391.1328 56 #> 6 6 382.6667 18.274527 6 337.9505 427.3828 67 #> 10 10 437.6667 18.274527 6 392.9505 482.3828 7
# Genetic variability analysis gva.augmentedRCBD(out1)
#> $Mean #> [1] 81.0625 #> #> $PV #> [1] 72.26786 #> #> $GV #> [1] 45.29563 #> #> $EV #> [1] 26.97222 #> #> $GCV #> [1] 8.302487 #> #> $`GCV category` #> [1] "Low" #> #> $PCV #> [1] 10.48703 #> #> $`PCV category` #> [1] "Medium" #> #> $ECV #> [1] 6.406759 #> #> $hBS #> [1] 62.67743 #> #> $`hBS category` #> [1] "High" #> #> $GA #> [1] 10.99216 #> #> $GAM #> [1] 13.5601 #> #> $`GAM category` #> [1] "Medium" #>
gva.augmentedRCBD(out2)
#> $Mean #> [1] 298.4792 #> #> $PV #> [1] 4980.411 #> #> $GV #> [1] 4694.161 #> #> $EV #> [1] 286.25 #> #> $GCV #> [1] 22.95435 #> #> $`GCV category` #> [1] "High" #> #> $PCV #> [1] 23.64387 #> #> $`PCV category` #> [1] "High" #> #> $ECV #> [1] 5.668377 #> #> $hBS #> [1] 94.25248 #> #> $`hBS category` #> [1] "High" #> #> $GA #> [1] 137.2223 #> #> $GAM #> [1] 45.97382 #> #> $`GAM category` #> [1] "High" #>